reserve n for Nat;

theorem Th9:
  for f be FinSequence st f is weakly-one-to-one holds Rev f is
  weakly-one-to-one
proof
  let f be FinSequence;
  assume
A1: f is weakly-one-to-one;
A2: len f = len Rev f by FINSEQ_5:def 3;
  let i be Nat;
  assume that
A3: 1 <= i and
A4: i < len Rev f;
A5: i+1 <= len Rev f by A4,NAT_1:13;
  i+1 >= 1 by NAT_1:11;
  then
A6: i+1 in Seg len Rev f by A5,FINSEQ_1:1;
  then i+1 in dom Rev f by FINSEQ_1:def 3;
  then
A7: Rev f.(i+1) = f.(len f - (i+1) + 1) by FINSEQ_5:def 3;
A8: i in Seg len Rev f by A3,A4,FINSEQ_1:1;
  then i in dom Rev f by FINSEQ_1:def 3;
  then
A9: Rev f.i = f.(len f - i + 1) by FINSEQ_5:def 3;
  len f - (i+1) + 1 = len f - i;
  then reconsider j = len f - i as Nat by A2,A5,FINSEQ_5:1;
  len f - i + 1 in Seg len f by A2,A8,FINSEQ_5:2;
  then j+1 <= len f by FINSEQ_1:1;
  then
A10: j < len f by NAT_1:13;
  len f - (i+1) + 1 in Seg len f by A2,A6,FINSEQ_5:2;
  then j >= 1 by FINSEQ_1:1;
  hence thesis by A1,A9,A7,A10;
end;
